[Solved] Nonlinear-control Homework 8

1. Consider the one-form = (1+2x₁+x₂)dx1 + x₁dx₂. Is it exact ? If yes, compute (x₁, x₂) such that =d(x₁, x₂).

2. Consider the one-form = x₁dx₂ + x₂dx₃ + x₃dx₄ . Is it integrable ? Compute d d .

Do there exist two functions ₁and ₂ such that span{d₁,d₂} ? If yes, compute ₁and ₂ .

3. The nonholonomic integrator

Check the accessibility of the following nonholonomic integrator:

x&1 =u1 x&2 =u2

x&3 =x1u 2 x2u1

Is it fully (means input-state) linearizable by static state feedback ?

Is it fully linearizable by dynamic state feedback ? Hint: try to find 2 output functions with relative degree 1 and such that the decoupling matrix has rank 1 only !